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Lazy Computation with Exact Real Numbers
 PROCEEDINGS OF THE THIRD ACM SIGPLAN INTERNATIONAL CONFERENCE ON FUNCTIONAL PROGRAMMING (ICFP98), VOLUME 34, 1 OF ACM SIGPLAN NOTICES
, 1997
"... We extend the framework for exact real arithmetic using linear fractional transformations from the nonnegative numbers to the extended real line. We then present an extension of PCF with a real type which introduces an eventually breadthfirst strategy for lazy evaluation of exact real numbers. In ..."
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Cited by 8 (3 self)
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We extend the framework for exact real arithmetic using linear fractional transformations from the nonnegative numbers to the extended real line. We then present an extension of PCF with a real type which introduces an eventually breadthfirst strategy for lazy evaluation of exact real numbers. In this language, we present the constant redundant if, rif, for defining functions by cases which, in contrast to parallel if (pif), overcomes the problem of undecidability of comparison of real numbers in finite time. We use the upper space of the onepoint compactification of the real line to develop a denotational semantics for the lazy evaluation of real programs. Finally two adequacy results are proved, one for programs containing rif and one for those not containing it. Our adequacy results in particular provide the proof of correctness of algorithms for computation of singlevalued elementary functions.
Streaming RepresentationChangers
 LNCS
, 2004
"... Unfolds generate data structures, and folds consume them. ..."
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Cited by 3 (0 self)
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Unfolds generate data structures, and folds consume them.
Incremental Addition in Exact Real Arithmetic
, 1998
"... Boehm et al. mention three different main approaches to exact real number arithmetic: Representation of reals via continued fractions, signed digit expansions, and as functions (Cauchy sequences). There exist prototype implementations of packages providing exact real arithmetic based on all three of ..."
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Boehm et al. mention three different main approaches to exact real number arithmetic: Representation of reals via continued fractions, signed digit expansions, and as functions (Cauchy sequences). There exist prototype implementations of packages providing exact real arithmetic based on all three of these approaches. A key property distinguishing the approaches is incrementality: If the accuracy of the result has to be increased in the function approach, computation starts from scratch and all previous calculations have to be disregarded. In contrast, the signed digit approach is incremental, i.e. the previous result is reused and some further digits are computed to increase precision. In this paper, we show how the function approach can be modified, resulting in a hybrid representation where signed digit expansions can be read as functions and vice versa. We develop an algorithm for addition in this setting combining advantages of both approaches. Keywords: Exact real arithmetic, in...
Exact Statistics and Continued Fractions
 Journal of Universal Computer Science
, 1995
"... Abstract: In this paper we investigate an extension to Vuillemin's work on continued ..."
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Abstract: In this paper we investigate an extension to Vuillemin's work on continued
Two Algorithms for Root Finding in Exact Real Arithmetic
, 1998
"... We present two algorithms for computing the root, or equivalently the fixed point, of a function in exact real arithmetic. The first algorithm uses the iteration of the expression tree representing the function in real arithmetic based on linear fractional transformations and exact floating point. T ..."
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We present two algorithms for computing the root, or equivalently the fixed point, of a function in exact real arithmetic. The first algorithm uses the iteration of the expression tree representing the function in real arithmetic based on linear fractional transformations and exact floating point. The second and more general algorithm is based on a trisection of intervals and can be compared with the wellknown bisection method in numerical analysis. It can be applied to any representation for exact real numbers; here it is described for the sign binary system in [\Gamma1; 1] which is equivalent to the exact floating point with linear fractional transformations. Keywords : Shrinking intervals, Normal products, Exact floating point, Expression trees, Sign Binary System, Iterative method, Trisection. 1 Introduction In the past few years, continued fractions and linear fractional transformations (lft), also called homographies or Mobius transformations, have been used to develop various...
Exact Real Number Computation Using Linear Fractional Transformations
"... which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy. ..."
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which has provided the first proper data type for solving ordinary differential equations up to any degree of accuracy.